Algebraic combinatorics, matrix integrals and algebraic geometry

组合学是研究有限对象集合的排列和它们之间的关系。组合学的研究主要是由其他数学和物理科学以及工程中的应用驱动的。许多这样的应用程序只需要确定集合中对象的数量，称为“计数”集合。在代数组合学中，我们广泛地应用了代数的结果和方法。最基本的组合对象之一被称为置换，它将有限集合中的元素重新排序。置换是一种非常简单的排列，它只是互换两个元素的顺序（“置换”）。排列的乘积是通过连续的重排序得到的，在代数中，一个集合的所有排列的乘积的集合称为对称群。

Combinatorics is the study of arrangements of sets of finite objects and relationships between them. Research in combinatorics is centrally motivated by applications in other mathematical and physical sciences, and in engineering. Many such applications require only that one determine the number of objects in a set, called "counting" the set. In algebraic combinatorics we make extensive use of results and methods from algebra in this study. One of the most fundamental combinatorial objects is called a permutation, which reorders the elements of a finite set. A transposition is a very simple permutation, that simply interchanges the order of ("transposes") two of the elements. A product of permutations is obtained by successive reordering, and in algebra, the set of all permutations of a set with this product is called the symmetric group. The purpose of this Research Proposal is to study applications of algebraic combinatorics to matrix integrals and algebraic geometry, to problems that have been especially significant in the recent research literature because of the surprising variety of areas in mathematics and physics that their solutions involve. For example, Hurwitz numbers arise in geometry as the number of ramified covers of the sphere, but this is equivalent in purely combinatorial terms to the number of ways in which a given permutation can be expressed as a product of transpositions in the symmetric group with a certain "connectivity" condition, and a given number of permutations in the product. These numbers and a new variant are studied in this proposal, and have been of great recent research interest because they have been significant in the study of 2-dimensional gravity, integrable hierarchies, matrix integrals, moduli spaces, enumerative geometry, and combinatorics. Of major interest in this proposal is to make the interaction between algebraic combinatorics and geometry two-way, not simply solving a geometric problem as stated, but to go further and obtain additional information about the original geometric questions themselves, or suggest new methods of solution within geometry.